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# Geometric theory of PDE and several complex variables

 Grant number: 12/03168-7 Support type: Research Projects - Thematic Grants Duration: November 01, 2012 - October 31, 2017 Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis Principal researcher: Jorge Guillermo Hounie Grantee: Jorge Guillermo Hounie Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil Pesquisadores principais: Adalberto Panobianco Bergamasco ; Gerson Petronilho ; Paulo Domingos Cordaro Assoc. researchers: Gustavo Hoepfner ; José Ruidival Soares dos Santos Filho ; Luís Antônio Carvalho dos Santos ; Paulo Leandro Dattori da Silva ; RAFAEL FERNANDO BAROSTICHI ; Sérgio Luís Zani Associated grant(s): Associated scholarship(s):

Abstract

The main purpose of the project is to continue the work undertaking by the research team of Projeto Temático 2003/12206-0 in the fields of Linear Partial Differential Equations and Multidimensional Complex Analysis as well as to increase our activities on supervision of graduate students research work in these areas. The main topics to be studied are: (a) Local, semi-global and global solvability for linear differential operators and involutive systems of complex vector fields; (b) Regularity properties of the solutions: $C^\infty$, analytic and Gevrey hypoellipticity; (c) General properties of the approximate solutions to involutive systems of complex vector fields; (d) The theory of Hardy spaces for solutions of non-elliptic vector fields; (e) The extension of the F. and M. Riesz and Rudin-Carleson theorems to complex vector fields. (AU)

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Scientific publications (23)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
BERGAMASCO, ADALBERTO P.; DATTORI DA SILVA, PAULO L.; GONZALEZ, RAFAEL B.. Existence of global solutions for a class of vector fields on the three-dimensional torus. BULLETIN DES SCIENCES MATHEMATIQUES, v. 148, p. 53-76, . (12/03168-7, 15/20815-4)
BAROSTICHI, RAFAEL F.; HIMONAS, A. ALEXANDROU; PETRONILHO, GERSON. Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems. JOURNAL OF FUNCTIONAL ANALYSIS, v. 270, n. 1, p. 330-358, . (12/00678-4, 12/03168-7)
HOEPFNER, G.; HOUNIE, J.; PICON, T.. Div-curl type estimates for elliptic systems of complex vector fields. Journal of Mathematical Analysis and Applications, v. 429, n. 2, p. 774-799, . (13/17636-5, 14/22054-8, 12/03168-7)
CERNIAUSKAS, WANDERLEY A.; DATTORI DA SILVA, PAULO L.. Solvability near the characteristic set for a class of first-order linear partial differential operators. Mathematische Nachrichten, v. 291, n. 8-9, p. 1240-1268, . (12/03168-7, 15/20815-4)
BERGAMASCO, ADALBERTO P.; DATTORI DA SILVA, PAULO L.; GONZALEZ, RAFAEL B.. Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, v. 23, n. 1, p. 65-90, . (12/03168-7)
BERGAMASCO, A. P.; DATTORI DA SILVA, P. L.; GONZALEZ, R. B.. Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus. Journal of Differential Equations, v. 264, n. 5, p. 3500-3526, . (12/03168-7, 15/20815-4)
CAMPANA, C.; DA SILVA, P. L. DATTORI; MEZIANI, A.. Riemann-Hilbert problem for a class of hypocomplex vector fields. Complex Variables and Elliptic Equations, v. 62, n. 10, SI, p. 1413-1424, . (13/26463-7, 13/08452-8, 14/06515-5, 12/03168-7)
BAROSTICHI, RAFAEL F.; HIMONAS, A. ALEXANDROU; PETRONILHO, GERSON. Global analyticity for a generalized Camassa-Holm equation and decay of the radius of spatial analyticity. Journal of Differential Equations, v. 263, n. 1, p. 732-764, . (12/03168-7)
CORDARO, PAULO D.; HOUNIE, JORGE. Semi-global solvability with loss of one derivative of partial differential operators on surfaces. ADVANCES IN MATHEMATICS, v. 301, p. 458-485, . (12/03168-7)
HIMONAS, A. ALEXANDROU; PETRONILHO, GERSON. EVOLUTION OF THE RADIUS OF SPATIAL ANALYTICITY FOR THE PERIODIC BBM EQUATION. Proceedings of the American Mathematical Society, v. 148, n. 7, p. 2953-2967, . (12/03168-7)
HOUNIE, JORGE; ZUGLIANI, GIULIANO. GLOBAL SOLVABILITY OF REAL ANALYTIC INVOLUTIVE SYSTEMS ON COMPACT MANIFOLDS. PART 2. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 371, n. 7, p. 5157-5178, . (14/23748-3, 12/03168-7)
ARAUJO, GABRIEL; CORDARO, PAULO D.. Real-analytic solvability for differential complexes associated to locally integrable structures. JOURNAL OF FUNCTIONAL ANALYSIS, v. 276, n. 2, p. 380-409, . (12/03168-7)
CORDARO, PAULO D.; JAHNKE, MAX REINHOLD. Top-Degree Global Solvability in CR and Locally Integrable Hypocomplex Structures. JOURNAL OF GEOMETRIC ANALYSIS, v. 31, n. 8, p. 8156-8172, . (12/03168-7)
PETRONILHO, GERSON; DA SILVA, PRISCILA LEAL. On the radius of spatial analyticity for the modified Kawahara equation on the line. Mathematische Nachrichten, v. 292, n. 9, p. 2032-2047, . (12/03168-7)
BERGAMASCO, ADALBERTO PANOBIANCO; PARMEGGIANI, ALBERTO; ZANI, SERGIO LUIS; ZUGLIANI, GIULIANO ANGELO. Geometrical proofs for the global solvability of systems. Mathematische Nachrichten, v. 291, n. 16, p. 2367-2380, . (12/05355-9, 12/03168-7)
BERHANU, S.; HOUNIE, JORGE. A HOPF LEMMA FOR HOLOMORPHIC FUNCTIONS IN HARDY SPACES AND APPLICATIONS TO CR MAPPINGS. JOURNAL D ANALYSE MATHEMATIQUE, v. 138, n. 2, p. 835-855, . (12/03168-7)
HOUNIE, JORGE; PICON, TIAGO. L-1 Sobolev estimates for (pseudo)-differential operators and applications. Mathematische Nachrichten, v. 289, n. 14-15, p. 1838-1854, . (13/17636-5, 12/03168-7)
DATTORI DA SILVA, P. L.; MEZIANI, A.. Cohomology relative to a system of closed forms on the torus. Mathematische Nachrichten, v. 289, n. 17-18, p. 2147-2158, . (14/06515-5, 12/03168-7)
CAMPANA, C.; DATTORI DA SILVA, P. L.; MEZIANI, A.. A class of planar vector fields with homogeneous singular points: Solvability and boundary value problems. Journal of Differential Equations, v. 265, n. 10, p. 5297-5314, . (12/03168-7, 16/21969-8, 13/08452-8, 15/20815-4)
BERGAMASCO, ADALBERTO; PARMEGGIANI, ALBERTO; ZANI, SERGIO; ZUGLIANI, GIULIANO. Classes of globally solvable involutive systems. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, v. 8, n. 4, p. 551-583, . (12/05355-9, 12/03168-7)
BERGAMASCO, ADALBERTO P.; DE MEDEIRA, CLEBER; KIRILOV, ALEXANDRE; ZANI, SERGIO L.. On the global solvability of involutive systems. Journal of Mathematical Analysis and Applications, v. 444, n. 1, p. 527-549, . (12/03168-7)
DE MEDEIRA, CLEBER; ZANI, SERGIO L.. A class of globally non-solvable involutive systems on the torus. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, v. 10, n. 2, p. 455-474, . (12/03168-7)
BAROSTICHI, R. F.; FERRA, I. A.; PETRONILHO, G.. Global hypoellipticity and simultaneous approximability in ultradifferentiable classes. Journal of Mathematical Analysis and Applications, v. 453, n. 1, p. 104-124, . (12/03168-7)